IMHO, you,ve still got carb probs (fuel delivery). The other things that you describe are normal for that engine.

Remove/disasm carb - soak overnite in carb cleaner - probe all orifi/passageways with a small dia stiff copper wire - flush and then blow out with compressed air - re-asm using the proper rebuild kit/instructions.

Thanks for the reply rcbe. I have had the carb off twice, but I admit I haven't probed all the passage ways. I have soaked it and blown it out with compressed air. It looks pretty clean. I have noticed that playing with the idle mixture wasn't making a whole lot of difference. That's why I got the new needle. I just put it on yesterday, but have not tried playing with the mixture with the new one. Once I get it started, it does idle with low throttle, but sounds louder than I remember. As I said, it stalls when I try to cut grass (load), it is drivable, and the blades do engage. When blades are engaged, idle does pick up and it sounds better.

Just to verify, you think the valves are behaving correctly, and magneto position sounds right.

One more symptom, when I pull the plug, it is always black with a carbon-like substance.

Your concern about the valves is based on the brief opening of the exhaust valve? What you are seeing there is the action of the compression release (ACR). This is only supposed to occur at cranking speeds (or lower) and when the engine RPM increases, the ACR is disabled.
Have you been noticing any black smoke out the exhaust when the engine is under a load?
Is the "black carbon like substance" shiny and sticky, or is it dull and powdery looking?
Shiny & sticky indicates oil consumption through the combustion chamber. This could be due to a worn engine or it could be due to a leaking head gasket (leak between combustion chamber and pushrod gallery) pressurizing the crankcase, which will force oil into the intake manifold.
If shiny and sticky, you need to do a "cylinder leak down test". If not equipped to do a leak down test, you need to pull the valve cover and watch the space down inside the pushrod gallery while the engine is running. Shine a powerful light in there and watch for regular puffs of compression blowing into the gallery from the combustion chamber.
If dull and powdery, the engine is getting too much fuel..........or too little air.

It's dull and powdery. So, I have symptoms of too much fule and not enough fuel. Black and podwery means too much fuel, but stalling under load means not enough, right?

I've been fighting this thing off and on for quite awhile. Could the black & powdery be due to me using carb spray to get it to start? Also, based on my symptoms, would you rule out bad timing or need for a new ignition module? Once it starts, it runs (but rough), but it can't cut grass. Could the timing be off? Or, could it be missing on some strokes?

It is unlikely that use of carb spray cleaner would be the culprit here.
Let's check something else before buy a shopping cart full of parts.
Do you have a "fine scale" pocket ruler graduated in 1/100ths of inch? Or possibly have a dial indicator (or know someone who does)?
Similar symptoms have come up here in the past on Kohler Command engines that ended up being a worn out camshaft.
Come up with one of the measuring devices mentioned above and measure carefully how far the rocker arm (at the pushrod) moves from fully down to fully up when you rotate the engine by hand. This will be measuring the "cam lobe lift height".
The Kohler manual for this engine gives the MINIMUM lift height as follows: Intake = 8.96mm or .353 inch. Exhaust = 9.14mm or .360 inch.
If either of the lobes fail to lift the MINIMUM spec, the cam is worn and needs replacing.
When camshafts wear out, the engine can't "breath" properly, and power plummets.

Go to the Kohler Engines website and download a free service manual.

Here is a link that might be useful: CV service manual

It will probably be next weekend before I have time to pull the valve cover again.

I was messing with it yesterday, and after I got it to start, I messed with the idle, and got it running a little smoother, and was able to drive it, turn on blades and run it through some low grass. It did struggle with blades engaged in the grass on an incline.

It did die a few times, sometimes with a backfire.

I was able to retstart it (without carb spray) but it took a few cranks. But I noticed that when I try to start it, sometimes the engine sounds like it misfires, and it actually rotates backwards very briefly. I am looking at the fan cover on the top. Seems like plug is firing while piston is trying to come up.